**On the Distribution of Balls in Urns**

如果有 *n* 個不同的球分配到 *r* 個不同的箱子中, 則可能的方法有 *r*^{n} 種.
假如考慮 *n* 個相同的球, 則可能的分法又如何呢?
以
來表示將 *n* 個相同的球分配到 *r* 個箱子的可能,
其中 *x*_{i} 代表在第 *i* 個箱子中的球數. 則此問題在解
之非負的整數解個數.

**Proposition 1**

There are
distinct **positive** integer-valued vectors
satisfying

**Proposition 2**

There are
distinct **nonnegative** integer-valued
vectors
satisfying

**Example**- An investor has 20 thousand dollars to invest among 4 possible investments.
Each investment must be in units of a thousand dolars. If the total 20 thousand
is to be invested, how many different investment strategies are possible? What
if not all the money need be invested?

*Solution:*- If we let
*x*_{i},*i*=1,2,3,4, denote the number of thousands invested in investment number*i*, then, when all is to be invested,*x*_{1},*x*_{2},*x*_{3},*x*_{4}are integers satisfying

Hence, by Proposition 2, there are possible investment strategies. If not all of the money need be invested, then, if we let*x*_{5}denote the amount kept in reserve, a strategy is a nonnegative integer-valued vector (*x*_{1},*x*_{2},*x*_{3},*x*_{4}) satisfying

Hence, by Proposition 2, there are now possible strategies.